Rare Entries XVII - Miscellaneous (history)

[Tube Geek]

But (at the moment) it isn't.

The definition of a prime number is 'a number that has 2 factors - itself and1'. But as for 1, itself and 1 are the same thing, it has only 1 factor, rather than 2. Therefore it is not a prime.

[The Orange One]

Actually (adapting the definition for rings), an integer p is prime if it satisfies the following:

1) p is not zero
2) p is not a unit - i.e. there is no integer q such that pq=1
3) If p divides ab, then p divides a or p divides b - where a and b are both integers.

A similar concept to primality is irreducibility. An integer p is irreducible if:

1) p is not zero
2) p is not a unit - i.e. there is no integer q such that pq=1
3) If p=ab, with a and b both integers, then either a or b is a unit.

So, under the definition:

• 0 is not prime (it is zero)
• 1 is not prime (it is a unit)
• 2 is prime
• -2 is prime
• 4 is not prime (4 divides 6 x 2 = 12, but does not divide either 6 or 2)

to name a few examples!

In the integers, primes and irreducibles are one and the same. In all rings, p is prime implies p is irreducible, but there are rings where p can be irreducible but not prime!

[tractakid]

Thanks for fixing this, by the way.

[A Challenge]

Yet I can find four factors for 2, thereby disrupting the construct of Prime Numbers:

1, 2, -1, -2 (1*2=2 and -1*-2=2)

[Tube Geek]

OK, using negative number means there are no prime numbers at all!

1x2=2, -1x-2=2
1x3=3, -1x-3=3
1x5=5, -1x-5=5
etc. etc.

[The Orange One]

No, that two factors rule is rubbish. I've told you, if it's not zero or a unit (1, -1) and the condition holds it's a prime. That's the definition.